Topological Data Analysis (TDA) applies the methods of Geometric Topology to the study of complex data. The subject is now about 15 years old, having gone through a striking evolution through basic theory, algorithm and software development, and applications to a variety of scientific problems. Recently, there has been a lot of work to bring TDA into a more effective position in both statistics and machine learning. In this talk we will review this history, and describe some new potential directions for TDA.

John Harer is Professor of Mathematics and Electrical and Computer Engineering at Duke University. His research focuses on three main areas: geometric and topological methods for data analysis, dynamics on networks, and mathematical biology. He is one of the founders of Topological Data Analysis, a new branch of science that sits between mathematics and computer science, and he has developed numerous algorithms and methods for finding shape features of objects and data based on topological approaches. His major results in this area include developing persistent homology and persistent local homology to study shape in high-dimensional datasets, establishing stability for persistence diagrams, and merging topology with statistics, diffusion geometry and machine learning. He and his research group are developing an extensive library of computational topology software that is easily integrated into other software systems. His book with Herbert Edelsbrunner is the primary reference for computational topology today. Professor Harer also works on both biological and other types of networks. He studies the problem of inferring network controls from topology and time series data. And he works on the design of networks that can self-heal and adapt to damage. Professor Harer has served as PI or Co-PI on numerous research grants from a variety of agencies including NSF, NIH, OSD, DARPA, AFOSR, DTRA and NIST. His small company, Geometric Data Analytics, focuses on application of geometric and topological data analysis to applied problems.